In DQN, when do the parameters in the Neural Network update based on the reward received?
However, all this is doing is updating the parameters of the Neural Network to produce an output that matches the Target Q values as close as possible.

Yes. That is all it needs to do because we have defined the policy around the Q values like so:
$$\pi(s) = \text{argmax}_a \hat{q}(s,a,\theta)$$
Where $\theta$ is the neural network weights.
Therefore, if the estimates of Q are approximately the same as the action value of the optimal policy, the policy in DQN is approximately the optimal policy.

How does the algorithm know +10 is good?

It does not, at least not directly. The algorithm knows, approximately, what the action values are if it acts consistently with their current estimates by always choosing the maximising action at each step.
The learning process will learn that +10 is relatively good in your scenario because it never finds anything better when exploring.